#6387601

order	512
nilpotency class	3
isoclinism	isoclinic class of #61569
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

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	C₂²×#449	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3002 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ci,d^b=df,g^c=gi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282214, #72282591
maximal normal subgroup of	#72282214, #72282591

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6294797);