#7562612

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518992
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	C₂²×#496	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2471 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)(7,11)(8,9)(10,12)(15,21)(16,23)(18,22)(19,25)(24,26)(27,28);(1,2)(3,4)(6,9)(10,11)(13,14,17,20)(15,22,24,28)(16,25,23,19)(18,26,27,21);(13,15,23,26,17,24,16,21)(14,18,25,28,20,27,19,22);(2,4)(13,15,23,26,17,24,16,21)(14,18,25,28,20,27,19,22);(5,7,8,12)(6,10,9,11)(15,24)(18,27)(21,26)(22,28)

Polycyclic presentation:

2^9|b^2=i,c^2=gi,d^2=gi,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^c=ei,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257084, #72257120, #72257158, …
maximal normal subgroup of	#72257084, #72257120, #72257158, …
maximal quotient of	#72264877, #72308265, #72320003

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7469808);