#7613824

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
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 ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2069 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251279, #72256885, #72256895, …
maximal normal subgroup of	#72251279, #72256885, #72256895, …
maximal quotient of	C₂×#7613824

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7521020);