#6606434

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574015
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#38049	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1799 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=g,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,d^c=dh,e^d=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264004, #72264011, #72264030
maximal normal subgroup of	#72264004, #72264011, #72264030
maximal quotient of	#72264586, #72264897

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6513630);