#498572

order	512
nilpotency class	4
isoclinism	isoclinic class of #33856
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#2246	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

710 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 710 conjugacy classes of subgroups — too many to draw. Only its 194 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2,4,3)(5,6,10,14,11,19,22,31,13,16,21,28,12,7,8,17)(9,15,24,29,25,32,34,36,27,30,33,35,26,18,20,23);(1,3,4,2)(5,7)(6,12)(8,17)(9,23)(10,28)(11,16)(13,19)(14,21)(15,20)(18,24)(22,31)(25,35)(26,29)(27,36)(30,34)(32,33);(5,8,12,21,13,22,11,10)(6,14,19,31,16,28,7,17)(9,20,26,33,27,34,25,24)(15,29,32,36,30,35,18,23);(5,9,8,20,12,26,21,33,13,27,22,34,11,25,10,24)(6,15,17,23,7,18,28,35,16,30,31,36,19,32,14,29)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=fi,d^2=eh,e^2=hi,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fi,d^b=de,e^b=eh,g^b=gh,h^b=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216402, #72216414, #72216416, …
maximal normal subgroup of	#72216402, #72216414, #72216416, …
maximal quotient of	#72226848

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405768);