#497360

order	512
nilpotency class	4
isoclinism	isoclinic class of #480866
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	#37653	C₂
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#547 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1216 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,d^b=de,g^c=gh,g^f=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312322, #72312349, #72312355, …
maximal normal subgroup of	#72312322, #72312349, #72312355, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404556);