#7021587

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996375
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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2448 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307857, #72308103, #72308193
maximal normal subgroup of	#72307857, #72308103, #72308193

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6928783);