#6685304

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682532
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2039 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:

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

Polycyclic presentation:

2^9|c^2=gi,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307879, #72307888, #72307963, …
maximal normal subgroup of	#72307879, #72307888, #72307963, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6592500);