#382392

order	512
nilpotency class	3
isoclinism	isoclinic class of #184805
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂²×#534	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

4650 conjugacy classes of subgroups; only the 362 normal subgroups are shown.

normal subgroups only

This group has 4650 conjugacy classes of subgroups — too many to draw. Only its 362 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) = 28
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|c^2=h,b^a=be,c^a=cf,d^a=di,h^a=hi,c^b=cg,h^b=hi,d^c=de,f^c=fi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72236931
maximal normal subgroup of	#72236931
maximal quotient of	#72236932, #72236970

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 289588);