#7612013

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518952
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	#59124	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2270 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,d^2=h,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,d^c=di,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282182, #72282988
maximal normal subgroup of	#72282182, #72282988

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7519209);