#494953

order	512
nilpotency class	3
isoclinism	isoclinic class of #188774
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#432
prime radical	C₂²×D₄	M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

570 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=e,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,d^b=dh,e^b=ei,g^d=gi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239422, #72252775, #72252778
maximal normal subgroup of	#72239422, #72252775, #72252778

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402149);