#6597042

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574031
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	#36097	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2402 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=fhi,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=eh,d^c=dh,e^d=eh

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307516, #72307546, #72308063
maximal normal subgroup of	#72307516, #72307546, #72308063

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6504238);