#493111

order	512
nilpotency class	2
isoclinism	isoclinic class of #7855
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×C₄²
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#97
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

788 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,e^2=i,g^2=h,h^2=i,b^a=be,d^a=di,g^a=gi,c^b=ch,f^b=fi,d^c=di,g^c=gi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319799, #72319812, D₄×#2110, …
maximal normal subgroup of	#72319799, #72319812, D₄×#2110, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400307);