#374129

order	512
nilpotency class	3
isoclinism	isoclinic class of #24472
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3021	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2487 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 20 points:

(1,2,3,4)(5,6)(7,17)(8,10)(9,19)(11,15)(12,16)(13,18)(14,20);(5,7)(6,13)(8,17)(9,16)(10,19)(11,18)(12,14)(15,20);(2,4)(5,8,11,16,12,15,10,6)(7,13,18,17,14,9,19,20);(5,9)(6,14)(7,16)(8,19)(10,17)(11,20)(12,13)(15,18)

Polycyclic presentation:

2^9|a^2=h,c^2=fhi,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^b=gi,g^c=gi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#24472, #72312219, #72312238
maximal normal subgroup of	C₄×#24472, #72312219, #72312238

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 281325);