#6393482

order	512
nilpotency class	3
isoclinism	isoclinic class of #62283
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3537	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3622 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=ghi,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^b=ei,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72292822, #72292824, #72292860
maximal normal subgroup of	#72292822, #72292824, #72292860
maximal quotient of	#72293220, #72293251

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6300678);