#7149033

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993979
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 ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2160 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)(7,11)(10,12)(13,14,17,22,18,21,15,19)(16,24,23,27,25,28,26,20);(1,2,3,4)(6,9)(7,11)(13,15,18,17)(14,19,21,22)(16,23,25,26)(20,24,27,28);(5,6)(7,10)(8,9)(11,12)(13,16)(14,20)(15,23)(17,26)(18,25)(19,24)(21,27)(22,28);(2,4)(6,9)(10,12)(13,15,18,17)(14,19,21,22)(16,23,25,26)(20,24,27,28);(5,7,8,11)(6,10,9,12)(14,21)(19,22)(20,27)(24,28)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307339, #72307388, #72308077
maximal normal subgroup of	#72307339, #72307388, #72308077

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7056229);