#6459741

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438949
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3252 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)(5,6)(7,9)(8,10)(11,12)(13,14)(15,23)(16,17)(18,21)(19,22)(20,26)(24,28)(25,27);(1,2,3,4)(5,7)(6,9)(8,11)(10,12)(13,15,17,26,19,25,18,24)(14,20,21,23,22,28,16,27);(6,10)(7,11)(13,16,17,14,19,21,18,22)(15,23,26,28,25,27,24,20);(7,11)(9,12)(13,17,19,18)(14,16,22,21)(15,24,25,26)(20,23,28,27);(15,25)(20,28)(23,27)(24,26)

Polycyclic presentation:

2^9|b^2=f,c^2=gh,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=ei,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216148
maximal normal subgroup of	#72216148
maximal quotient of	#72216058

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6366937);