#7503298

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35563	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2886 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)(8,10)(9,12)(13,14,19,25,18,16,17,21)(15,20,26,28,23,22,24,27);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,22)(17,24)(18,23)(19,26)(21,27)(25,28);(2,4)(13,16,18,14)(15,22,23,20)(17,21,19,25)(24,27,26,28);(6,10)(7,12)(13,14,18,16)(15,20,23,22)(17,25,19,21)(24,28,26,27);(15,23)(20,22)(24,26)(27,28)

Polycyclic presentation:

2^9|a^2=gi,c^2=i,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,e^b=ei,g^c=gi,g^d=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308676, #72308686, #72308847
maximal normal subgroup of	#72308676, #72308686, #72308847

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7410494);