#415502

order	512
nilpotency class	3
isoclinism	isoclinic class of #395174
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2487
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

776 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=e,c^2=e,d^2=h,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,f^b=fi,d^c=di,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249841, #72249842, #72249848, …
maximal normal subgroup of	#72249841, #72249842, #72249848, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322698);