#192103

order	512
nilpotency class	3
isoclinism	isoclinic class of #18192
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1262 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=hi,d^2=hi,e^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,d^b=dh,e^b=eh,f^b=fi,e^c=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 99299);