#6545183

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535664
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2378 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)(13,14)(15,22)(16,20)(17,27)(18,21)(19,25)(23,24)(26,28);(5,6)(7,10)(8,9)(11,12)(13,15,17,25,18,26,16,24)(14,19,20,22,21,23,27,28);(2,4)(6,9)(10,12)(13,14)(15,23)(16,27)(17,20)(18,21)(19,26)(22,24)(25,28);(5,7)(6,10)(8,11)(9,12)(13,16,18,17)(14,20,21,27)(15,24,26,25)(19,22,23,28);(7,11)(10,12)

Polycyclic presentation:

2^9|a^2=g,b^2=f,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^c=di,e^d=ehi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256928, #72256935, #72256951, …
maximal normal subgroup of	#72256928, #72256935, #72256951, …
maximal quotient of	#72265684

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6452379);