#497298

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	C₂×#547	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#547 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1171 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

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,d^c=di,g^c=gh,g^f=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312236, #72312245, #72312266, …
maximal normal subgroup of	#72312236, #72312245, #72312266, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404494);